\(\int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx\) [131]

Optimal result

Integrand size = 26, antiderivative size = 534 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{800 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{96 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2} \]

[Out]

-1/8*(d*x+c)^(3/2)*cos(b*x+a)/b-1/48*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b+1/80*(d*x+c)^(3/2)*cos(5*b*x+5*a)/b+3/8000
*d^(3/2)*cos(5*a-5*b*c/d)*FresnelS(b^(1/2)*10^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*10^(1/2)*Pi^(1/2)/b^(5/2)+
3/8000*d^(3/2)*FresnelC(b^(1/2)*10^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(5*a-5*b*c/d)*10^(1/2)*Pi^(1/2)/b^
(5/2)-1/576*d^(3/2)*cos(3*a-3*b*c/d)*FresnelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*6^(1/2)*Pi^(1/2)
/b^(5/2)-1/576*d^(3/2)*FresnelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)*6^(1/2)*Pi^(1
/2)/b^(5/2)-3/32*d^(3/2)*cos(a-b*c/d)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2
)/b^(5/2)-3/32*d^(3/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*2^(1/2)*Pi^(1/2)/
b^(5/2)+3/16*d*sin(b*x+a)*(d*x+c)^(1/2)/b^2+1/96*d*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b^2-3/800*d*sin(5*b*x+5*a)*(d*
x+c)^(1/2)/b^2

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b} \]

[In]

Int[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]

[Out]

-1/8*((c + d*x)^(3/2)*Cos[a + b*x])/b - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(48*b) + ((c + d*x)^(3/2)*Cos[5*a +
5*b*x])/(80*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])
/(16*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])
/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[
d]])/(800*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5
*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (
3*b*c)/d])/(96*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (
b*c)/d])/(16*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(96*b^2
) - (3*d*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(800*b^2)

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} (c+d x)^{3/2} \sin (a+b x)+\frac {1}{16} (c+d x)^{3/2} \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^{3/2} \sin (5 a+5 b x)\right ) \, dx \\ & = \frac {1}{16} \int (c+d x)^{3/2} \sin (3 a+3 b x) \, dx-\frac {1}{16} \int (c+d x)^{3/2} \sin (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^{3/2} \sin (a+b x) \, dx \\ & = -\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}-\frac {(3 d) \int \sqrt {c+d x} \cos (5 a+5 b x) \, dx}{160 b}+\frac {d \int \sqrt {c+d x} \cos (3 a+3 b x) \, dx}{32 b}+\frac {(3 d) \int \sqrt {c+d x} \cos (a+b x) \, dx}{16 b} \\ & = -\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}+\frac {\left (3 d^2\right ) \int \frac {\sin (5 a+5 b x)}{\sqrt {c+d x}} \, dx}{1600 b^2}-\frac {d^2 \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{192 b^2}-\frac {\left (3 d^2\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}+\frac {\left (3 d^2 \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {5 b c}{d}+5 b x\right )}{\sqrt {c+d x}} \, dx}{1600 b^2}-\frac {\left (d^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{192 b^2}-\frac {\left (3 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{32 b^2}+\frac {\left (3 d^2 \sin \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {5 b c}{d}+5 b x\right )}{\sqrt {c+d x}} \, dx}{1600 b^2}-\frac {\left (d^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{192 b^2}-\frac {\left (3 d^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}+\frac {\left (3 d \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{800 b^2}-\frac {\left (d \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{96 b^2}-\frac {\left (3 d \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^2}+\frac {\left (3 d \sin \left (5 a-\frac {5 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{800 b^2}-\frac {\left (d \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{96 b^2}-\frac {\left (3 d \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{800 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{96 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.54 (sec) , antiderivative size = 1126, normalized size of antiderivative = 2.11 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {d} e^{-\frac {3 i (a d+b (c+d x))}{d}} \left (-12 \sqrt {b} \sqrt {d} e^{\frac {3 i b c}{d}} \sqrt {c+d x} \left (-i+2 b x+e^{6 i (a+b x)} (i+2 b x)\right )-(1-i) (2 b c+i d) e^{\frac {3 i b (2 c+d x)}{d}} \sqrt {6 \pi } \text {erf}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+(1+i) (2 i b c+d) e^{3 i (2 a+b x)} \sqrt {6 \pi } \text {erfi}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{2304 b^{5/2}}-\frac {\sqrt {d} e^{-\frac {5 i (a d+b (c+d x))}{d}} \left (-20 \sqrt {b} \sqrt {d} e^{\frac {5 i b c}{d}} \sqrt {c+d x} \left (-3 i+10 b x+e^{10 i (a+b x)} (3 i+10 b x)\right )-(1-i) (10 b c+3 i d) e^{\frac {5 i b (2 c+d x)}{d}} \sqrt {10 \pi } \text {erf}\left (\frac {(1+i) \sqrt {\frac {5}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+(1+i) (10 i b c+3 d) e^{5 i (2 a+b x)} \sqrt {10 \pi } \text {erfi}\left (\frac {(1+i) \sqrt {\frac {5}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{32000 b^{5/2}}+\frac {i c d e^{-\frac {i (b c+a d)}{d}} \left (-e^{2 i a} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )+e^{\frac {2 i b c}{d}} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )\right )}{16 b^2 \sqrt {c+d x}}+\frac {1}{16} c \left (-\frac {e^{3 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},-\frac {3 i b (c+d x)}{d}\right )}{6 \sqrt {3} b \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{-3 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},\frac {3 i b (c+d x)}{d}\right )}{6 \sqrt {3} b \sqrt {\frac {i b (c+d x)}{d}}}\right )-\frac {1}{16} c \left (-\frac {e^{5 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},-\frac {5 i b (c+d x)}{d}\right )}{10 \sqrt {5} b \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{-5 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},\frac {5 i b (c+d x)}{d}\right )}{10 \sqrt {5} b \sqrt {\frac {i b (c+d x)}{d}}}\right )+\frac {\sqrt {d} \left (e^{i \left (a-\frac {b c}{d}\right )} \left (-2 \sqrt {b} \sqrt {d} e^{\frac {i b (c+d x)}{d}} (3 i+2 b x) \sqrt {c+d x}+\sqrt [4]{-1} (2 i b c+3 d) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )+i \left (2 \sqrt {b} \sqrt {d} (3+2 i b x) \sqrt {c+d x}+(1+i) (2 b c+3 i d) \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {2} \sqrt {d}}\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right ) (\cos (a+b x)-i \sin (a+b x))\right )}{64 b^{5/2}} \]

[In]

Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]

[Out]

(Sqrt[d]*(-12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(-I + 2*b*x + E^((6*I)*(a + b*x))*(I + 2*b*x)) -
(1 - I)*(2*b*c + I*d)*E^(((3*I)*b*(2*c + d*x))/d)*Sqrt[6*Pi]*Erf[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sq
rt[d]] + (1 + I)*((2*I)*b*c + d)*E^((3*I)*(2*a + b*x))*Sqrt[6*Pi]*Erfi[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x
])/Sqrt[d]]))/(2304*b^(5/2)*E^(((3*I)*(a*d + b*(c + d*x)))/d)) - (Sqrt[d]*(-20*Sqrt[b]*Sqrt[d]*E^(((5*I)*b*c)/
d)*Sqrt[c + d*x]*(-3*I + 10*b*x + E^((10*I)*(a + b*x))*(3*I + 10*b*x)) - (1 - I)*(10*b*c + (3*I)*d)*E^(((5*I)*
b*(2*c + d*x))/d)*Sqrt[10*Pi]*Erf[((1 + I)*Sqrt[5/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + (1 + I)*((10*I)*b*c + 3
*d)*E^((5*I)*(2*a + b*x))*Sqrt[10*Pi]*Erfi[((1 + I)*Sqrt[5/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/(32000*b^(5/2)
*E^(((5*I)*(a*d + b*(c + d*x)))/d)) + ((I/16)*c*d*(-(E^((2*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((-I)*b
*(c + d*x))/d]) + E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, (I*b*(c + d*x))/d]))/(b^2*E^((I*(b*c +
a*d))/d)*Sqrt[c + d*x]) + (c*(-1/6*(E^((3*I)*(a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2, ((-3*I)*b*(c + d*x))/d])/
(Sqrt[3]*b*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma[3/2, ((3*I)*b*(c + d*x))/d])/(6*Sqrt[3]*b*E^((3*
I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/16 - (c*(-1/10*(E^((5*I)*(a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2,
((-5*I)*b*(c + d*x))/d])/(Sqrt[5]*b*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma[3/2, ((5*I)*b*(c + d*x)
)/d])/(10*Sqrt[5]*b*E^((5*I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/16 + (Sqrt[d]*(E^(I*(a - (b*c)/d))*(-2*
Sqrt[b]*Sqrt[d]*E^((I*b*(c + d*x))/d)*(3*I + 2*b*x)*Sqrt[c + d*x] + (-1)^(1/4)*((2*I)*b*c + 3*d)*Sqrt[Pi]*Erfi
[((-1)^(1/4)*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]) + I*(2*Sqrt[b]*Sqrt[d]*(3 + (2*I)*b*x)*Sqrt[c + d*x] + (1 + I)*(
2*b*c + (3*I)*d)*Sqrt[Pi/2]*Erf[((1 + I)*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[2]*Sqrt[d])]*(Cos[b*(c/d + x)] + I*Sin[b
*(c/d + x)]))*(Cos[a + b*x] - I*Sin[a + b*x])))/(64*b^(5/2))

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{8 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{16 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{10 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{100 b \sqrt {\frac {b}{d}}}\right )}{80 b}}{d}\)	\(580\)
default	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{8 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{16 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{10 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{100 b \sqrt {\frac {b}{d}}}\right )}{80 b}}{d}\)	\(580\)

[In]

int((d*x+c)^(3/2)*cos(b*x+a)^2*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2/d*(-1/16/b*d*(d*x+c)^(3/2)*cos(b/d*(d*x+c)+(a*d-b*c)/d)+3/16/b*d*(1/2/b*d*(d*x+c)^(1/2)*sin(b/d*(d*x+c)+(a*d
-b*c)/d)-1/4/b*d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos((a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c
)^(1/2)/d)+sin((a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))-1/96/b*d*(d*x+c)^(3/2)*
cos(3*b/d*(d*x+c)+3*(a*d-b*c)/d)+1/32/b*d*(1/6/b*d*(d*x+c)^(1/2)*sin(3*b/d*(d*x+c)+3*(a*d-b*c)/d)-1/36/b*d*2^(
1/2)*Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^
(1/2)/d)+sin(3*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))+1/160/b*d*(d*x+
c)^(3/2)*cos(5*b/d*(d*x+c)+5*(a*d-b*c)/d)-3/160/b*d*(1/10/b*d*(d*x+c)^(1/2)*sin(5*b/d*(d*x+c)+5*(a*d-b*c)/d)-1
/100/b*d*2^(1/2)*Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*(cos(5*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2
)*b*(d*x+c)^(1/2)/d)+sin(5*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.80 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (30 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{5} - 50 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{3} - {\left (9 \, b d \cos \left (b x + a\right )^{4} - 13 \, b d \cos \left (b x + a\right )^{2} - 26 \, b d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{72000 \, b^{3}} \]

[In]

integrate((d*x+c)^(3/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/72000*(27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi
*d))) - 125*sqrt(6)*pi*d^2*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d
))) - 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))
- 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) -
125*sqrt(6)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 27
*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) + 480
*(30*(b^2*d*x + b^2*c)*cos(b*x + a)^5 - 50*(b^2*d*x + b^2*c)*cos(b*x + a)^3 - (9*b*d*cos(b*x + a)^4 - 13*b*d*c
os(b*x + a)^2 - 26*b*d)*sin(b*x + a))*sqrt(d*x + c))/b^3

Sympy [F]

\[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**(3/2)*cos(b*x+a)**2*sin(b*x+a)**3,x)

[Out]

Integral((c + d*x)**(3/2)*sin(a + b*x)**3*cos(a + b*x)**2, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.42 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(3/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/288000*sqrt(2)*(1800*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(5*((d*x + c)*b - b*c + a*d)/d)/d^2 - 3000*sqrt(2)*(d*x
+ c)^(3/2)*b^4*cos(3*((d*x + c)*b - b*c + a*d)/d)/d^2 - 18000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(((d*x + c)*b - b
*c + a*d)/d)/d^2 - 540*sqrt(2)*sqrt(d*x + c)*b^3*sin(5*((d*x + c)*b - b*c + a*d)/d)/d + 1500*sqrt(2)*sqrt(d*x
+ c)*b^3*sin(3*((d*x + c)*b - b*c + a*d)/d)/d + 27000*sqrt(2)*sqrt(d*x + c)*b^3*sin(((d*x + c)*b - b*c + a*d)/
d)/d + 27*((I + 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) - (I - 1)*25^(1/4)*sqrt(pi)*b^2
*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) + 125*(-(I + 1)*9^(1/4)*sqrt(pi)*b^2*
(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (I - 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*er
f(sqrt(d*x + c)*sqrt(3*I*b/d)) + 6750*(-(I + 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (I - 1)*sqr
t(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + 6750*((I - 1)*sqrt(pi)*b^2*(b^
2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I + 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)
*sqrt(-I*b/d)) + 125*((I - 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) - (I + 1)*9^(1/4)*sqr
t(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)) + 27*(-(I - 1)*25^(1/4)*sqr
t(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) + (I + 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c -
a*d)/d))*erf(sqrt(d*x + c)*sqrt(-5*I*b/d)))*d^2/b^5

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 2314, normalized size of antiderivative = 4.33 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(3/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/144000*(300*(30*sqrt(2)*sqrt(pi)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e
^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) + 5*sqrt(6)*sqrt(pi)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d
)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) -
3*sqrt(10)*sqrt(pi)*d*erf(-1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-5*(I*b*c -
I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + 30*sqrt(2)*sqrt(pi)*d*erf(-1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x
+ c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + 5*sqrt(6)*sqr
t(pi)*d*erf(1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(-I*b*c + I*a*d)/d)/(sqr
t(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) - 3*sqrt(10)*sqrt(pi)*d*erf(1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/
sqrt(b^2*d^2) + 1)/d)*e^(-5*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)))*c^2 + d^2*(2250*(sqrt(
2)*sqrt(pi)*(4*b^2*c^2 + 4*I*b*c*d - 3*d^2)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2)
+ 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 2*I*(2*I*(d*x + c)^(3/2)*b*d - 4*I*
sqrt(d*x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + 125*(sqrt(6)*sqrt
(pi)*(12*b^2*c^2 - 4*I*b*c*d - d^2)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*
e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 6*I*(2*I*(d*x + c)^(3/2)*b*d - 4*I*sqrt(d
*x + c)*b*c*d - sqrt(d*x + c)*d^2)*e^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 - 9*(sqrt(10)*sqrt(pi)*(
100*b^2*c^2 - 20*I*b*c*d - 3*d^2)*d*erf(-1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e
^(-5*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 10*I*(-10*I*(d*x + c)^(3/2)*b*d + 20*I*sqr
t(d*x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^(-5*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + 2250*(sqrt(2)*sqr
t(pi)*(4*b^2*c^2 - 4*I*b*c*d - 3*d^2)*d*erf(-1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d
)*e^((-I*b*c + I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 2*I*(2*I*(d*x + c)^(3/2)*b*d - 4*I*sqrt(d
*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2 + 125*(sqrt(6)*sqrt(pi)*(1
2*b^2*c^2 + 4*I*b*c*d - d^2)*d*erf(1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(
-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 6*I*(2*I*(d*x + c)^(3/2)*b*d - 4*I*sqrt(d*x +
c)*b*c*d + sqrt(d*x + c)*d^2)*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2 - 9*(sqrt(10)*sqrt(pi)*(100*b^
2*c^2 + 20*I*b*c*d - 3*d^2)*d*erf(1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-5*(
-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 10*I*(-10*I*(d*x + c)^(3/2)*b*d + 20*I*sqrt(d*
x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^(-5*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2) - 20*(450*sqrt(2)*sqrt(
pi)*(2*b*c + I*d)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)
/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 25*sqrt(6)*sqrt(pi)*(6*b*c - I*d)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d
)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b)
- 9*sqrt(10)*sqrt(pi)*(10*b*c - I*d)*d*erf(-1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d
)*e^(-5*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) + 450*sqrt(2)*sqrt(pi)*(2*b*c - I*d)*d*erf(
-1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqrt(b*d)*(I*b*d/s
qrt(b^2*d^2) + 1)*b) + 25*sqrt(6)*sqrt(pi)*(6*b*c + I*d)*d*erf(1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/s
qrt(b^2*d^2) + 1)/d)*e^(-3*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 9*sqrt(10)*sqrt(pi)*
(10*b*c + I*d)*d*erf(1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-5*(-I*b*c + I*a*
d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 900*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b +
150*sqrt(d*x + c)*d*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 90*sqrt(d*x + c)*d*e^(-5*(I*(d*x + c)*b - I*
b*c + I*a*d)/d)/b + 900*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b + 150*sqrt(d*x + c)*d*e^(-3*(
-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b - 90*sqrt(d*x + c)*d*e^(-5*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b)*c)/d

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]

[In]

int(cos(a + b*x)^2*sin(a + b*x)^3*(c + d*x)^(3/2),x)

[Out]

int(cos(a + b*x)^2*sin(a + b*x)^3*(c + d*x)^(3/2), x)